Dimension formulas for period spaces via motives and species

TL;DR

This paper derives explicit dimension formulas for spaces of period numbers using motives and algebraic structures, generalizing classical results and confirming conjectures related to multiple zeta values.

Contribution

It introduces a new approach using finite dimensional algebra theory to compute dimensions of period spaces, extending classical and recent results.

Findings

Complete dimension formulas for 1-periods.

Generalization of Baker's theorem on logarithms.

Dimension estimates for multiple zeta values.

Abstract

We apply the structure theory of finite dimensional algebras in order to deduce dimension formulas for spaces of period numbers, i.e., complex numbers defined by integrals of algebraic nature. We get a complete and conceptually clear answer in the case of $1$-periods, generalising classical results like Baker's theorem on the logarithms of algebraic numbers and partial results in Huber--W{\"u}stholz \cite{huber-wuestholz}. The application to the case of Mixed Tate Motives (i.e., Multiple Zeta Values) recovers the dimension estimates of Deligne--Goncharov \cite{deligne-goncharov}.